with rational [resp. integer] coefficients. Thus the coefficients of GGG are symmetric polynomialsgjsubscriptgjg_{j} in the numbers α1(i)superscriptsubscriptα1i\alpha_{1}^{{(i)}}:

G=∑jgj⁢xjGsubscriptjsubscriptgjsuperscriptxjG\;=\;\sum_{j}g_{j}x^{j}

By the fundamental theorem of symmetric polynomials, the coefficients gjsubscriptgjg_{j} of GGG are polynomials in
α2,…,αksubscriptα2normal-…subscriptαk\alpha_{2},\,\ldots,\,\alpha_{k} with rational [resp. integer] coefficients. Consequently, GGG has the form

where the coefficients ai′⁢(α2,…,αk)superscriptsubscriptainormal-′subscriptα2normal-…subscriptαka_{i}^{{\prime}}(\alpha_{2},\,\ldots,\,\alpha_{k}) are polynoms in the numbers αjsubscriptαj\alpha_{j} with rational [resp. integer] coefficients. As one continues similarly, removing one by one also α2,…,αksubscriptα2normal-…subscriptαk\alpha_{2},\,\ldots,\,\alpha_{k} which go back to the rational [resp. integer] coefficients of the corresponding minimal polynomials, one shall finally arrive at an equation

among the roots of which there are the roots of (1); the coefficients AνsubscriptAνA_{\nu} do no more explicitely depend on the algebraic numbers α1,…,αksubscriptα1normal-…subscriptαk\alpha_{1},\,\ldots,\,\alpha_{k} but are rational numbers [resp. integers].
Accordingly, the roots of (1) are algebraic numbers [resp. algebraic integers], Q.E.D.